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3.Trigonometrical Ratios, Functions and Identities
medium
The value of $6({\sin ^6}\theta + {\cos ^6}\theta ) - 9({\sin ^4}\theta + {\cos ^4}\theta ) + 4$ is
A
$-3$
B
$0$
C
$1$
D
$3$
Solution
(c) $6({\sin ^6}\theta + {\cos ^6}\theta ) – 9({\sin ^4}\theta + {\cos ^4}\theta ) + 4$
$ = 6[{({\sin ^2}\theta + {\cos ^2}\theta )^3} – 3{\sin ^2}\theta {\cos ^2}\theta ({\sin ^2}\theta + {\cos ^2}\theta )]$
$ – 9[{({\sin ^2}\theta + {\cos ^2}\theta )^2} – 2{\sin ^2}\theta {\cos ^2}] + 4$
$ = 6[1 – 3{\sin ^2}\theta {\cos ^2}\theta ] – 9\,[1 – 2{\sin ^2}\theta {\cos ^2}\theta ] + 4$
$ = 6 – 9 + 4 = 1$.
Standard 11
Mathematics
